Download Nonlinear-optical properties of a noninteracting Bose gas

May 1, 1996 / Vol. 21, No. 9 / OPTICS LETTERS
677
Nonlinear-optical properties of a noninteracting Bose gas
M. A. Kasevich
Department of Physics, Stanford University, Stanford, California 94305
S. E. Harris
Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305
Received October 30, 1995
We show that the characteristic Rabi frequency and nonlinear susceptibility of atoms in a dilute Bose gas
remain unchanged by the process of condensation. We neglect the effects of atomic dipole – dipole interactions
and spontaneous emission. © 1996 Optical Society of America
The recent field-opening experiments of Anderson
et al.1 and Bradley et al.2 have demonstrated the
formation of a Bose –Einstein condensate of atoms
whose mean spacing is small compared with their
de Broglie wavelength and large compared with the
optical wavelength of their resonance transition. Because the de Broglie wavelength is larger than the
average distance between the atoms, individual atoms
cannot be distinguished, and the condensate must
be considered a single, macroscopic, quantum state.
When one does so, one finds that the matrix element
for optical excitation between the symmetrized Bose
product states varies as the square root of the number
of particles in the condensate.
One might therefore be led to anticipate such extraordinary macroscopic properties as (i) a Rabi frequency
that itself varies as the square root of the number of
particles or, because third-order susceptibilities vary
as the fourth power of the matrix element, (ii) a thirdorder susceptibility that is proportional to the square
of the number of atoms in the sample.
In this Letter we show that, if all interactions between particles are neglected, these properties do not
exist and, in fact, that the Rabi frequency and the nonlinear susceptibility of the sample are unchanged by
condensation. In our model we assume N degenerate two-state atoms that interact with a semiclassical
radiation field. We ignore spontaneous emission and
atomic dipole – dipole interactions and consider the spatial extent of the condensate to be much larger than
the optical wavelength. Hiroshima and Yamamoto3 recently calculated x !3" for ensembles of tightly conf ined
degenerate Bose and Fermi atoms. You and coworkers4 considered the optical response of confined Bose
atoms under the conditions of short-pulse excitation.
Other related studies have focused on the linear-optical
response of a Bose gas.5 – 8
To show the interesting and perhaps peculiar way
in which the condensate properties reduce to those
of single atoms, we work the problem in the atomic
number state basis. As illustrated in Fig. 1, we let
state jq# denote the excitation of q of these atoms
from their internal ground state ja# to their excited
state jb#. For simplicity we take the center-of-mass
state for atoms in ja# to be the momentum p ! 0
eigenstate and that of atoms in state jb# to be the
0146-9592/96/090677-03$10.00/0
p ! h̄k momentum eigenstate (k is the wave vector of
the incident electromagnetic f ield). The states jq# are
then eigenstates of the free particle Hamiltonian H0
with eigenenergies
∂
µ
h̄2 k2
H0 jq# ! q h̄v0 1
jq# ,
(1)
2m
where h̄v0 is the internal electronic excitation energy
and h̄2 k2 $2m is the single-photon recoil energy.
The ith particle in the condensate interacts with an
electric field E!Ri " ! E $2 exp% j!vt 2 k ? Ri "& 1Pc.c.
N
through the electric – dipole interaction V ! 2e i!1
ri ? E!Ri ". The interaction matrix element is
'qjV jq 1 1# ! 2
h̄V
!q 1 1"1/2 !N 2 q"1/2 exp! jvt" 1 c.c. ,
2
(2)
where the single-atom Rabi frequency V ( mab ? E $
h̄ ! mab is the single-atom dipole matrix element). The
scaling of the matrix element with N and q follows
directly from the required exchange symmetry of the
condensate wave functions. Note that in the limit
N .. q the strength of the interaction matrix element
scales as N 1/2 .
We study the dynamics of the light –condensate
interaction in the interaction picture. With jc!t"# !
P
N
and the rotating-wave
q!0 aq !t"exp!2jqvt"jq#
Fig. 1. (a) For a total of N atoms, in the basis of bare,
independent atoms there are 2N possible basis states; (b) in
the symmetrized Bose basis there are (N 1 1) states. The
qth state is that where q atoms are excited.
© 1996 Optical Society of America
678
OPTICS LETTERS / Vol. 21, No. 9 / May 1, 1996
approximation, we arrive at the following coupled
equations describing the evolution of the probability
amplitudes aq !t":
≠aq
V!
V
1 jqDvaq ! j
jq21 aq21 1 j
jq aq11
≠t
2
2
!0 # q # N " ,
X
ri d!R 2 Ri " .
'P !3" !R"# ! 2
1 mab jVj2 V
exp% j!vt 2 k ? R"&
V
4Dv 3
3 !j0 4 2 j0 2 j1 2 $2" 1 c.c.
(3)
where jq ! !q 1 1"1/2 !N 2 q"1/2 , the detuning Dv !
v0 1 h̄k2 $2m 2 v, and the spurious amplitudes a21
and aN 11 are zero for all t. Taking N ! 1, Eq. (3)
reduces to the standard pair of coupled equations for
a two-level atom interacting with a driving radiation
field. On resonance, these equations lead to the wellknown sinusoidal oscillation of population between the
ground and excited states at the Rabi frequency V.
We first investigate the limit N .. 1 by numerically integrating Eq. (3) for large numbers of
atoms.9 Figure 2(a) shows the on-resonance time
evolution of the probability of f inding an ensemble of
N ! 100 atoms in the j0# state subject to the initial condition that, at t ! 0, all the atoms are in state ja#. We
find what at f irst is strange behavior: the population
is rapidly depleted at a rate proportional to N 1/2 V but
then returns on a time scale set by the single-atom Rabi
frequency V.
To gain insight into the collective evolution of the entire set of probability amplitudes we show in Fig. 2(b)
the time evolution
of the mean number of excited
P
atoms, 'q# ! qjaq j2 . Here we f ind that the observable 'q# oscillates at frequency V in exactly the same
manner as a collection of N nondegenerate atoms.
As an example of a second observable for this system,
we consider the induced polarization operator P!R":
P !R" ! e
the optical f ield is turned on we f ind that
!2
(6a)
N mab jVj2 V
exp%j!vt 2 k ? R"& 1 c.c.
V
4Dv 3
(6b)
The f irst term in Eq. (6a) is the contribution of perturbation path a [shown in Fig. 3(a)]; the second is that of
path b [shown in Fig. 3(b)]. Note that either path, if
alone, yields a contribution proportional to N 2 . When
both paths are included, the N 2 terms cancel and 'P !3" #
is the same as that of N nondegenerate atoms.
From these results one is led to wonder whether all
stimulated optical properties of a dilute condensate
reduce to those of a nondegenerate gas. Further
insight can be gained by consideration
of the class of
P
Hamiltonians of the form H ! i H !i", where the
operators H !i" depend on the coordinates of the ith
particle alone. For example, the Hamiltonian for
the noninteracting Bose gas considered above falls
into this class, but one that includes atom–atom
interactions does not. For this class of Hamiltonians the equations of motion are separable in the
coordinates of the individual particles,
and the
Q
condensate wave function jc!t"# ! i jwi !t"#, where
jwi !t"# are solutions to the single-particle equations
H !i" jwi !t"# ! j h̄≠$≠tjwi !t"#. [Note that jc!t"# is automatically symmetric under particle interchange if all
particles initially reside in the same quantum state.]
We now specialize this result to the example of
Rabi oscillations considered above. The on-resonance,
(4a)
i
Assuming box normalization to volume V of the
momentum eigenstates, the matrix elements of the
dipole operator are
'qjP !R"jq 1 1# !
jq
mab exp! j k ? R" ,
V
(4b)
and the expected value of the induced polarization
'P!R"# is
1
mab exp% j !vt 2 k ? R"&
'c!t"jP !R"jc!t"# !
V
N
21
X
jq aq aq11 ! 1 c.c.
3
Fig. 2. Numerical integration of the coupled equations
[Eq. (3)] for N ! 100 atoms: (a) population in state j0#,
(b) mean number of excited atoms 'q#.
(5)
q!0
In the limit where the detuning Dv is much larger
than the Rabi frequency jVj, one can use perturbation theory to calculate 'P# to arbitrary order in the
small parameter jVj$Dv. The third-order term in
this expansion, 'P !3" #, determines the third-order susceptibility x !3" and sets the magnitude of such effects
as self-focusing and self-phase modulation. With the
standard assumption that the condensate adiabatically
evolves from the lowest-lying state j0# to state jc!t"# as
Fig. 3. Perturbation paths for calculation of x !3" :
a, (b) path b.
(a) path
May 1, 1996 / Vol. 21, No. 9 / OPTICS LETTERS
single-atom solutions are jci # ! cos!Vt$2"jai # 1
sin!Vt$2"jbi #. Here the kets jai # and jbi # correspond
to the ground state (with momentum p ! 0) and the
excited state (with momentum p ! h̄k), respectively, of
the ith particle. In this Q
notation the number state j0#
of the N atom system is i jai #. Thus the probability
of f inding an atom in state j0# [as shown in Fig. 2(a)]
is cos2N !Vt$2". We also find immediately that 'q#
[shown in Fig. 2(b)] oscillates at frequency V. Similar
arguments apply to calculation of the induced
dipole
P
moment. It follows directly that 'P # ! i 'Pi #, where
'Pi # are the single-atom contributions to the induced
dipole moment.
In conclusion, we have shown that the Rabi frequency and susceptibility do not depend on the momentum degeneracy of dilute atomic samples and that these
results follow directly from the separability of the collective Hamiltonian.
The authors acknowledge helpful discussions with J.
Cooper, E. Cornell, H. Haus, H. Hioe, A. Kapitulnik,
and Y. Yamamoto. This research was supported by
679
the U.S. Army Research Office, the U.S. Air Force
Office of Scientific Research, the National Science
Foundation, and the U.S. Off ice of Naval Research.
References
1. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.
Wieman, and E. A. Cornell, Science 269, 198 (1995).
2. C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.
Hulet, Phys. Rev. Lett. 75, 1687 (1995).
3. T. Hiroshima and Y. Yamamoto, ‘‘Nonlinear optical
response of cold atoms,’’ submitted to Phys. Rev. A.
4. L. You, M. Lewenstein, and J. Cooper, Phys. Rev. A 51,
4712 (1995).
5. O. Morice, Y. Castin, and J. Dalibard, Phys. Rev. A 51,
3896 (1995).
6. H. Politzer, Phys. Rev. A 43, 6444 (1991).
7. J. Javanainen, Phys. Rev. Lett. 72, 2375 (1994).
8. L. You, M. Lewenstein, and J. Cooper, Phys. Rev. A 50,
R3565 (1994).
9. Analytic solutions to Eq. (3) appear in F. Hioe, J. Opt.
Soc. Am. B 4, 1327 (1987).